**1. One to one function:**

A function *f* that assigns every element of the domain A to exactly one element of the codomain B is called a one to one function.

Note:

It is not necessary for every element of the codomain B to be assigned to some element of the range of f, set A for the mapping f to be a one to one function.

*This condition exists for a function to be onto as below. *

**2. Onto function: **

A mapping*g* that assigns every element of the codomain B to some element of the range A is called an onto function.

g is an onto function in the figure above because every element of the codomain B is assigned in the function.

Note that in the above function, two elements 1 and 4 are mapped to a same element c in the codomain B and the function is still an onto.

Therefore, a function is onto even though some elements of the codomain may be mapped to distinct elements of the domain of f, i.e. an onto function need not be one to one.

IMPORTANT CONDITION FOR A FUNCTION TO BE ONTO:

*For a function to be onto, none of the elements of the codomain B must be left unmapped. *

The function below is ** not**an onto function, because one element in the codomain B,

Therefore,

*in an onto function, codomain = range of f.*

i.e. the two sets codomain and range of the function f have the same elements and of course the same number of elements too.

**3. Bijective Function: **

A function *f* that is both onto and one to one is called a Bijective function.

Since, the function is one to one, therefore no element of the codomain must be mapped to more than one element of the domain A and also since it is onto, therefore none of the elements of the codomain must be left unmapped.

**4. Identity function: **

A mapping f that associates every element of the domain to itself is called an identity function.

An identity function f is denoted as:

**5. Constant Function: **

A mapping that maps every element of the domain to only one element in the codomain is called a constant function.

In the above function f from A to B, every element of the domain of f, i.e set A is mapped to one and only one element of the codomain B.

Therefore the above mapping f from A to B is called a constant function.

Note that the range of a constant function has only one element.

Recall that a set that contains only one element is called a singleton set.

Therefore, the range of a constant function is a singleton set.

The range of a constant function f can be denoted as {k}, where k is the element in the codomain to which all of the elements of the domain f are mapped.

**6. Algebraic functions: **

A function consisting of a finite number of terms of the independent variable having powers and roots of it and the four fundamental arithmetic operations of addition, subtraction, multiplication and division is called an algebraic function.

*f(x) = 2x + 3, *

*g(x) = x ^{2} + 5x + 6*

*h(x) = √(x + 1)*

*k(x) = (5x + 6)/(3x – 8)*

Algebraic functions are divided into three types. They are:

**Polynomial function or rational integral function **

**Rational function****Irrational function**

**7. Transcendental functions**

Functions which are not algebraic functions are called transcendental functions.

Some of the transcendental functions are:

**Trigonometric functions****Inverse trigonometric functions.****Exponential functions****Logarithmic functions**

**8. Explicit function**

A mapping is said to be an explicit function when the dependent variable is exclusively, i.e. explicitly, a function of only the dependent variable.

**Example: **

*f(x) = x ^{2} – 2x + 1. *

In this example, the dependent variable *y*, i.e. *f(x), *varies only as x changes, i.e. it depends only on the dependent variable x, and not also on itself, like in implicit function as seen below.

**9. Implicit function**

A mapping is said to be an implicit function when the dependent variable*y*varies according to both the independent variable, denoted by *x*, and also the dependent variable, denoted by *y*

**Example: **

*f(x) = x ^{2} – y^{2}*

In this example, the dependent variable *y*, i.e. *f(x), varies not only as x does, but also as y, itself, does. *

**10. Even function**

A function** f** which is such that

**Example 1: **

f(x) = x^{2} is an even function, because

*f(–x) = (-x) ^{2 }= x^{2} = f(x)*

**Example 2: **

f(x) = cosx is also an even function, because

f(-x) = cos(-x) = cosx = f(x)

**11. Odd function**

A function** f** which is such that

**Example 1: **

f(x) = x^{3} is an even function, because

*f(–x) = (-x) ^{3 }= –x^{3} = –f(x)*

**Example 2: **

f(x) =sinx is also an odd function, because

f(-x) = sin(-x) = -sinx = -f(x)

**12. Greatest integer function or step function**

A step function or more famously called as the greatest integer function is defined as below:

*f(x) = n*, where n is the greatest integer less than or equal to n.

Denotation of greatest integer function:

The greatest integer function is denoted by the symbol [x].

Where **[x] = n ≤ x**, where n, according to the definition, is the greatest integer less than or equal to x.

The following examples will make clear the greatest integer function.

**Example 1: **

[4.5] = 4, since 4 is the greatest integer which is less than 4.5

**Example 2: **

[5] = 5, since 5 is the greatest integer which is equal to 5.

**Example 3: **

[─5.5] = ─6, since -6 is the greatest integer which is less than ─5.5

**13. Modulus function or Absolute value function. **

The function y = │x│ is called modulus function or absolute value function.

Now │x│= x, if x ≥ 0 and │x│=─x, if x < 0

**Example: **

│3│ = 3, since 3> 0 and

│─4│=─(─4) = 4, since ─4< 0

**14. Inverse function**

Let f be a function from A to B. then, the inverse of the function f is a function from B to A.

The inverse of the function ** f** is denoted by

Note:

The inverse of a function ** f**will exist i.e. it will be a function if and only if the function f is one to one and onto.

First of all in the function **f: A – B**, A is domain of f and B is the range of f.

Then, the inverse of f denoted by ** f-1**will be a function from B to A, i.e. denoted as

Now, if the function f is not one to one, then the domain B of will reduce to a one to many mapping which is against the definition of a function and again if the function is not onto, then in the domain B of ** f-1**there will be some elements which are not mapped to some element in the range of

How to find the inverse of a function f:

Let f(x) = x + 2. Then, what is the inverse of f?

**Solution: **

First of all, put y = f(x), then, from this, we get *x = f-1(y)*

Now, given that y = x + 2, so, x = y – 2

So, *f-1(y) = y – 2 *

Now, just replace the variable y with x, and find the inverse of f(x) as below:

*f-1(x) = x – 2*

**15. Signum function**

**16. Undefined or Indeterminate functions **

**17. Composite function or product function **

**18. Real value function **

**19. Real variable function**

**20. the square root function **

**21. the reciprocal function **

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